DISCUSSION PAPER SERIES
No. 6429
FISCAL POLICY, LABOUR UNIONS
AND MONETARY INSTITUTIONS:
THEIR LONG RUN IMPACT ON
UNEMPLOYMENT, INFLATION
AND WELFARE
Alex Cukierman and Alberto Dalmazzo
INTERNATIONAL MACROECONOMICS,
LABOUR ECONOMICS and PUBLIC
POLICY
ABCD
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FISCAL POLICY, LABOUR UNIONS
AND MONETARY INSTITUTIONS:
THEIR LONG RUN IMPACT ON
UNEMPLOYMENT, INFLATION
AND WELFARE
Alex Cukierman, Tel Aviv University and CEPR
Alberto Dalmazzo, University of Siena
Discussion Paper No. 6429
August 2007
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Copyright: Alex Cukierman and Alberto Dalmazzo
CEPR Discussion Paper No. 6429
August 2007
ABSTRACT
Fiscal Policy, Labour Unions and Monetary Institutions: Their Long
Run Impact on Unemployment, Inflation and Welfare*
Objectives and motivation: This paper considers the impact of interactions
between fiscal policy and monetary institutions in the presence of unionized
labour markets on economic outcomes and welfare in the long run. Two main
classes of questions are investigated. First, what is the impact of exogenously
given labour taxes and unemployment benefits on the choice of monetary
policy by the central bank, on the choice of nominal wages by unions, on the
choice of prices by monopolistically competitive firms and through them on
unemployment, inflation and welfare? Second, how are labour taxes and
redistribution chosen by a (Stackelberg leader) fiscal authority whose
objectives are a weighted average of social welfare and of catering to the
interests of political supporters, and how does the general equilibrium induced
by this choice affect welfare? The framework of the paper is motivated by the
European scene in which the fraction of the labour force covered by collective
agreements dominates wage setting in the labour market.
"Players” and payoffs: The model economy features labour unions that
maximize the expected real income of union members over states of
employment and of unemployment, a central bank that strives to minimize the
combined costs of inflation and of unemployment, and a continuum of
monopolistically competitive firms, each of which maximizes its profits. The
last part of the paper also features a fiscal authority that sets taxes and
redistribution so as to maximize a combination of social welfare and of
benefits to particular constituencies. Utility from consumption is characterized
by means of a CES, Dixit-Stiglitz, utility function and (as in Sidrauski type
models) money appears in the utility function.
Methodology and “players” strategies: The first question is investigated within
a three stage game in which labour unions move first and commit to nominal
wages and the central bank moves second and chooses the money supply. In
the third and last stage each of a large number of monopolistically competitive
firms picks its price. To deal with the second class of questions the game is
expanded to feature a preliminary stage in which government chooses labour
taxes and redistribution anticipating the subsequent responses of the other
players. General equilibrium is characterized and used to find the impact of
various economic and institutional parameters.
JEL Classification: E5, E6, H2, J3, J5 and L1
Keywords: fiscal monetary policy interactions, labour unions, monopolistic
competition, monetary institutions, long run inflation unemployment tradeoff,
welfare, politics and fiscal policy
Alex Cukierman
Eitan Berglas School of Economics
Tel-Aviv University
Tel Aviv 69978
ISRAEL
Email: alexcuk@post.tau.ac.il
Alberto Dalmazzo
Dipartimento di Economia Politica
Facoltà di Economia
Piazza S. Francesco 7
I-53100 Siena
ITALY
Email: dalmazzo@unisi.it
For further Discussion Papers by this author see:
For further Discussion Papers by this author see:
www.cepr.org/pubs/new-dps/dplist.asp?authorid=102723
www.cepr.org/pubs/new-dps/dplist.asp?authorid=149123
*An earlier version of the paper was presented at the June 2007 Kiel
Symposium on "The Phillips Curve and the Natural Rate of Unemployment".
Submitted 07 August 2007
1
Introduction
This paper presents a general equilibrium microfounded macro framework characterized by imperfectly competitive goods and labor markets with sticky nominal wages and uses it to investigate the impact of economic structure and of fiscal and monetary policies on economic
performance in the long run. The framework of the paper is motivated by the European scene
in which the fraction of the labor force covered by collective agreements is large and in which
the degree of centralization of wage setting institutions varies across different countries.1
An important ingredient of our long run analysis, emphasized in some recent literature,
is that, in the presence of large wage setters, monetary policymaking institutions, like central
bank conservativeness (CBC) affect inflation as well as real variables even in the long run.2 As
a consequence the long run values of both unemployment and inflation are affected by the
level of CBC. The basic reason for this is that, when choosing its nominal (contractually set)
wage, a large wage setter internalizes the subsequent policy response of the monetary authority
to its own action. Since the nature of this response depends on CBC, nominal as well as real
wages, and the natural rate of unemployment depend on CBC. This mechanism does not appear
in most, now canonical, New-Keynesian macro models because those models are populated by
small price or wage setters that take monetary policy and other aggregate macroeconomic
variables as given.3 The paper does share with the New Keynesian literature the notion that
1
In many European countries union membership is at least fifty percent. But, even in those countries in which
membership is lower than fifty percent the fraction of the labor force actually covered by collective bargaining
is substantially higher. This is achieved through various legal and other institutionalized extensions of union
wages to segments of the labor force that are not members of unions. According to OECD (1997) coverage rates
ranged from 68% in Spain to 92% in France at the beginning of the nineties.
2
A non exhaustive list includes Skott (1977), Cukierman and Lippi (1999), Soskice and Iversen (2000), Lippi
(2003) and Coricelli, Cukierman and Dalmazzo (2006).
3
For example the labor market in chapter 3 of Woodford (2003) is assumed to be competitive. Although
Erceg, Henderson and Levin (2000) deviate from this paradigm by modeling labor suppliers as monopolistically
competitive agents, those agents do not believe that their individual actions affect the choice of monetary policy
or other aggregate variables. One possible reason for these choices of modeling strategy is the fact that, in the
US, the fraction of the labor force covered by collective agreements is substantially smaller than in Europe. A
3
some nominal variables are sticky. But, unlike some early papers in this literature, it explores
the view that wages are substantially stickier than prices.4
Although, it is an important ingredient of our analysis the above mentioned mechanism is
only one of several mechanisms that serve as a starting point for the main issues that constitute
the focus of this paper. The paper discusses both positive and normative issues. The positive
questions concern the effects of economic structure and of policy instruments and institutions
on inflation, unemployment and the functional distribution of income in the long run. Economic
structure is characterized by the degree of competitiveness in the good’s markets and by the
degree of centralization of wage bargaining (CWB). Policy instruments and institutions are
characterized by labor taxes, unemployment benefits and CBC. The normative issues focus on
the impact of economic structure and policy instruments and institutions on economic welfare.
In comparison to the past, central banks in developed economies now enjoy high levels of
both legal and actual independence (or equivalently, effective conservativeness) and are expected
and encouraged to conduct policy mainly to achieve price stability. They possess, therefore,
high levels of effective CBC in Rogoff (1985) sense. It would appear, therefore, that monetary
policy is independent of politically motivated decisions in the fiscal area. The paper argues that
such a view is oversimplistic. Although direct political interferences with monetary policy are
rare, fiscal policy decisions regarding labor taxes and redistribution affect the monetary policy
decisions of even highly conservative central banks provided those banks are concerned, at least
to some extent, with real economic activity.5
Intuitively this occurs because fiscal parameters affect both inflation and unemployment,
recent attempt to introduce unions into a New-Keynesian framework appears in Gnocchi (2005).
4
For example King and Wolman (1999) assume that prices are sticky and that wages are fully flexible.
Friedman (1999) critisizes this assumption as being unrealistic. Our paper makes a step towards meeting this
critisism while maintaining some analytical simplicity by exploring the diametrically opposite case in which wages
are contractually fixed and prices are flexible.
5
That is provided they are flexible inflation targeters in Svensson (1997) sense.
4
and because the central bank (CB) cares about the values of those variables. The paper explores
the channels through which fiscal parameters like labor taxes and unemployment benefits affect
monetary policy decisions and through them economic performance. In addition, although
fiscal policymakers are taken as first movers with respect to the CB, the policy response they
expect from the bank affects their decisions. The first part of the paper abstracts from this
reverse causality by assuming that fiscal policy settings are exogenous. The latter part takes
it up by endogenizing fiscal policy. Together, those discussions provide a general equilibrium
analysis of the long run impact of fiscal and monetary policies on the economy while taking into
consideration the interactions between them.6
Following is a brief description of the analytical building blocks of the paper. The model
economy features a number of non atomistic labor unions each of which maximizes the expected
real income of a representative union member over states of employment and unemployment,
a central bank that strives to minimize the combined costs of inflation and of unemployment,
and a continuum of monopolistically competitive firms, each of which maximizes its individual
profits.
For given fiscal policy instruments the investigation is conducted within a three stage
game in which labor unions move first and commit to contractually fixed nominal wages and
the central bank moves second and chooses the money supply. In the third and last stage each
monopolistically competitive firm picks its price. The equilibrium outcomes are then used to
derive the implications for long run inflation, unemployment, income distribution and related
macroeconomic variables. This framework is also used to evaluate the impact of economic structure and policymaking institutions on welfare. To deal with the impact of monetary institutions
on the choice of fiscal instruments, the game is expanded to feature a preliminary stage in
which government chooses labor taxes and unemployment benefits, taking into consideration
6
A first attempt to model the positive impact of those interactions in a somewhat different framework appears
in Cukierman and Dalmazzo (2006).
5
the subsequent responses of the other players to its own actions.
Methodologically, the paper provides a unified treatment of three distinct strands of recent literature. One on strategic interactions between labor unions and the CB; another, mostly
empirical, on the effects of labor taxes on unemployment and competitiveness in the presence of
unionized labor markets; and a third on interactions between fiscal and monetary policies. Some
of the references in the first group have been mentionned above (see note 2). A recent exhaustive
empirical examination of the impact of fiscal and labor market institutions on unemployment in
the OECD countries appears in Nickell, Nunziata and Ochel (2005).7 Interactions between fiscal
and monetary policies under commitment and discretion within a single country are investigated
by Dixit and Lambertini (2003).8 An important difference between the last paper and ours is
that it posits (at least implicitly) a competitive labor market while our model features a small
number of unions making it possible to characterize the three ways strategic interaction among
fiscal policymakers, labor unions and the CB.9
Section 2 provides a general overview of the main building blocks of the model economy.
Section 3 characterizes general equilibrium of the economy for given fiscal policy instruments.
In particular it derives consumer’s demand functions, prices set by firms, the money supply chosen by the CB, contractually fixed nominal wages set by labor unions and basic macroeconomic
variables like unemployment and inflation. Section 4 derives implications for real marginal costs,
the markup and the functional distribution of income, and section 5 summarizes general equilibrium comparative statics results. Section 6 discusses the effects of economic structure, fiscal
7
Related work includes Alesina and Perotti (1997), Daveri and Tabellini (2000) and Belot and van Ours
(2001).
8
Dixit (2001) and Dixit and Lambertini (2001) extend this analysis to the case of a monetary union.
9
Another difference is that they explore alternative combinations of regimes (discretion versus commitment)
for fiscal and monetary policies. By contrast, we consider a four stage game in which the fiscal authority moves
first, labor unions move second, the central bank sets monetary policy in the third stage and individual prices
are set by monopolistically competitive firms in the last stage. The motivation for those timing assumptions is
discussed in subsection 2.1 below
6
policy parameters and CBC on welfare. To this point all sections take fiscal policy instruments
as given. Section 7 endogenizes the choice of taxes and redistribution by adding a preliminary
stage to the previous three stages game. In this stage a government that cares about both redistribution and social welfare chooses taxes, taking into consideration the subsequent responses
of unions, the CB and price setters. This is followed by concluding remarks.
2
An overview of the model
There is a continuum of monopolistically competitive firms, whose mass is one, and n equally
sized labor unions that organize the entire labor force. Each firm is owned by an entrepeneur
whose income consists of the firm’s profits. The mass of firms in the economy is one. Each
monopolistic union covers the labor force of a fraction 1/n of the firms. A quantity L0 of
workers is attached to each firm. Without loss of generality, all firms whose labor force is
represented by union i are assigned to the contiguous subinterval ( ni , i+1
) of the unit interval,
n
where i = 0, 1..., n − 1.10 Utility of an individual in the economy is given by:
¶1−γ
µ ¶γ µ
M/P
C
+ (1 − λ)R, γ ∈ (0, 1)
U=
γ
1−γ
(1)
Here C is a Dixit-Stiglitz (1977) consumption aggregator
C=
µZ
1
θ−1
θ
Cj
dj
0
θ
¶ θ−1
,
θ>1
(2)
of imperfectly substituable consumption varieties, Cj , whose mass is equal to one and θ is the
(constant across varieties) elasticity of substitution between any pair of varieties. M denotes
the nominal money stock held by the individual, and the price level P (equal conceptually to
10
This part of the the model builds on Coricelli, Cukierman and Dalmazzo (2006).
7
the minimum cost of providing the utility level C) is given by:
P =
µZ
1
Pj1−θ
0
1
¶ 1−θ
(3)
where Pj is the price of variety j. A worker can be either employed or unemployed. R denotes
the difference between his utility from leisure when unemployed and employed so that λ = 0
when he is unemployed and λ = 1 when he is employed. Since all firm owners (entrepeneurs)
must forego leisure to manage their firms λ = 1 for each entrepeneur.11 Each individual, whether
a worker or a capitalist, possesses an initial endowment of money, M. This endowment is the
same for all individuals.
Let Acs denote total nominal resources available to individual s in class c where c =
EW, U W, E. Here EW, UW, E stand for "employed worker", "unemployed worker", and "employer" respectively. The budget constraint of individual s states that the nominal resources at
his disposition are used to satisfy his consumption demands for the different varieties, Ccsj , plus
his demand for nominal money balances, Mcs . It is given by
Acs = Mcs +
Z
1
Pj Ccsj dj, s = EW, UW, E
(4)
0
where
AEW s = WEW s + M + T RW , AUW s = B + M + T RW , As = Πs + M + T RE .
(5)
Here WEW s is the net nominal wage earned by employed worker s, B ≥ 0 is an unemployment
benefit paid by government to each unemployed worker, Πs is the profit received by firm owner
s, and T RW ,and T RE are governmental transfers to each worker and employer respectively.
11
This part of the model utilizes some of the structure from Blanchard and Kiyotaki (1987) and chapter 8 of
Blanchard and Fischer (1988).
8
For simplicity the transfers to each worker and employers are assumed to be the same across
workers and employers respectively. Each individual, s, whether worker or entrepeneur, chooses
consumption varieties, Ccsj , j ∈ [0, 1] and nominal money balances, Mcs , so as to maximize
utility, (1), subject to the budget constraint (4). In the rest of the paper we drop the individual
index, s, whenever there is no risk of confusion.
Government raises taxes on labor and utilizes the proceeds to finance unemployment
benefits, as well as other transfers. As in Alesina and Perotti (1997), there are two types of
taxes. A social security tax paid by the employer (at rate σ), and an income tax (at rate υ).
Denoting by Wg the gross wage paid to an employee, a firm bears a per-worker cost of labor
equal to (1 + σ)Wg , while the worker receives a net wage equal to W = (1 − ν)Wg .12 Thus, the
ratio between the net wage and the cost of labor to the firm is given by
cost of labor to the firm can be written as
W
.
(1−t)
1−ν
1+σ
≡ (1 − t), and the
Taking (natural) logarithms the last equation,
can be reformulated as log W − log(1 − t) ≡ w + τ , where − log(1 − t) ≡ τ > 0.
Given taxation, the real profits of firm j, whose workforce belongs to union i are given
by
Πij
Wi
Pj
= Cja −
Lij ,
P
P
P (1 − t)
j ∈ [0, 1]
(6)
where Cja denotes the aggregate demand for variety j. Taking the nominal wage, Wi , and the
general price level, P , as given each firm chooses the price, Pj , of the variety it sells so as to
maximize profits subject to the production function
Yj = Lαij ,
α < 1.
(7)
Yj is the amount of this variety that is produced and Lij is the number of workers employed by
12
Since the discussion here is a generic one we omit the indices of the particular worker and firm.
9
firm j.
Monetary institutions are represented by a central bank (CB) that dislikes both inflation,
π, and unemployment, u. As in Coricelli, Cukierman and Dalmazzo (2006), the CB chooses the
money supply so to minimize the combined costs of inflation and of unemployment that are
given by
Γ = u2 + I · π 2 , I ∈ [0, ∞).
(8)
As in Rogoff (1985), the parameter I measures the relative importance that the CB assigns to
the objective of low inflation versus low unemployment. This parameter is also known as the
degree of CB conservativeness (CBC).
The probability that a member of union i will be unemployed is identical and independent
across the union’s members so that the probability that any union member is unemployed is
equal to the rate of unemployment among its members. Taking the nominal wages of other
unions as given, each union, i, sets the nominal wage, Wi , for its members so as to maximize
the expected utility of a representative member. This expected utility is given by
vi = (1 − ui )VEW (AEW ) + ui VUW (AU W )
(9)
where ui is the unemployment rate among union i’s members and, VEW (.) and VU W (.) are
the individually maximal values of utility of employed and unemployed workers expressed as
functions of the resources available to each type of worker.
We assume that the "replacement ratio"
B
P
B
Wi
is constant and equal to β < 1 so that
= β WPi . We also postulate that the level of maximized utility when employed is larger than
the level of maximized utility when unemployed at all real wages above than or equal to the
competitive one so that
VEW (AEW ) ≥ VUW (AUW ).
10
(10)
As we shall see later this implies that all unemployment is involuntary. Hence if offered a job,
an unemployed worker will always accept it. We therefore refer to the constraint in (10) as a
"participation constraint".13
We turn next to government’s budget constraint. Government outlays are used to finance unemployment benefits, as well as other transfers. Since B denotes the nominal value
of the benefit to each unemployed, the total amount of unemployment benefits paid out by
the government is equal to uL0 B, where Lo is the mass of labor per firm.14 Government also
pays out the transfer T RW to each worker and the transfer T RE to each employer. Since the
mass of employers is one the total outlays for those transfers are Lo T RW + T RE . Denoting by
χ ≥ 0 the amount of such transfers as measured per-worker, the government pays out χL0 where
χ ≡ T RW +
1
T RE .
Lo
Government revenues come from employment taxes. Total tax revenues
are given by tWg (1 − u)L0 =
tW
(1
1−t
− u)L0 . We assume the budget is balanced implying that:
tW
(1 − u) = Bu + χ.
1−t
2.1
(11)
Timing
We postulate the following sequence of events. In the first stage government sets the fiscal
policy parameters. In the second stage each union chooses its nominal wage so as to maximize
its objective function (9). When doing this, the union takes the nominal wages set by other
unions as given, and anticipates the reactions of both the CB and the firms to its wage choice.
In the third stage the Central Bank chooses the nominal stock of money so as to minimize its
loss function (8), taking as given the preset nominal wages and anticipating the reaction of firms
to its choice. In the last stage each firm takes the general price level as given and sets its own
13
Strictly speaking it is enough to assume that the participation constraint holds only at the competitive real
wage since this assumption implies, a fortiori, that it also holds at higher real wages.
14
Since the mass of firms is one the aggregate mass of labor and the mass of labor per firm are identical.
11
price so as to maximize real profits.
This timing sequence is meant to capture, within a static model, the fact that nominal
wages are stickier than prices and that they normally are set for a period that is longer than the
period for which monetary policy is set.15 Note that, since there are no shocks in the model the
relative position of monetary policy and of price setting by firms within this timing sequence
is immaterial for the nature of equilibrium. The reason is that, in the absence of shocks firms
perfectly anticipate the subsequent choice of monetary policy by the CB. Hence they set the same
prices as those they would have set when monetary policy precedes price setting - - leading to the
same monetary policy and an identical equilibrium. We view fiscal authorities as the first mover
because changes in tax rates and unemployment benefits require new legislation by Parliament.
This is usually a lenghty process that requires more time than changes in monetary policy.
General equilibrium is characterized by backward induction. We start by solving the
firms’ pricing problem, then the CB problem, and finally the unions’ nominal wage decisions.
The first part of the paper characterizes equilibrium in the last three stages for given values
of transfer payments, unemployment benefits and taxes. In the second part we discuss the
choice of fiscal policy parameters by a (partially) politically motivated government in stage 1.
In particular, we discuss the impact of redistributive policies in favor of "special interests" on
the nature of equilibrium.
3
General equilibrium
General equilibrium is characterized by backward induction. First, the price choice of each firm,
given nominal wages and the money supply, is derived. Second the choice of the money supply
by the CB, given nominal wages, is characterized. Finally, the choice of nominal wage by each
15
For analytical simplicity and in order to focus on the implications of the higher relative stickiness of nominal
wages we abstract from the fact that some prices are sticky too.
12
union is calculated. Characterization of price setting decisions by each firm requires knowledge
of the demand facing each firm. Since this demand depends on the behavior of consumers we
start with the maximization problem of a typical consumer.
3.1
The individual consumer.
Each individual maximizes utility (1) with respect to each consumption variety, Cj , j ∈ [0, 1]
and money, M, subject to (4). In the symmetric general equilibrium developed later in the
paper all individuals within a given class, c = EW, UW, E possess identical demands for each
of the consumption goods and money. It therefore suffices to characterize those demands for a
representative individual within each class. It is shown in Appendix 9.1 that the demands of an
individual from class c for variety j and for money balances that evolve from maximization of
utility are given respectively by
Ccj = γ
µ
Pj
P
¶−θ µ
Ac
P
¶
, j ∈ [0, 1], c = EW, UW, E
Mc = (1 − γ)Ac , c = EW, U W, E
(12)
(13)
and that the indirect utility function of the representative individual within each class is given
by
vc =
Ac
+ (1 − λ)R, c = EW, UW, E
P
(14)
where λ = 0 for c = UW and λ = 1 otherwise. Equation (14) in conjunction with equation (5)
implies that the indirect utility functions of each of the three types of individuals (employed,
13
unemployed and capitalists) are given respectively by
W + M + T RW
≡ AEW ,
P
B + M + T RW
+ R ≡ AUW ,
=
P
Π + M + T RE
≡ AE .
=
P
vEW =
vUW
vE
3.2
(15)
Aggregate equilibrium conditions and the demand facing an individual firm
From (13) aggregate demand for money is equal to:
Mad = (1 − γ) [AEW + AUW + AE ] ≡ (1 − γ)Aa .
Using the definition χ ≡ T RW +
1
T RE
Lo
(16)
and government’s budget constraint in (11), total
demand for money may be rewritten as
Mad
∙
µ
= (1 − γ) (1 + L0 )M + (1 − u)L0 W 1 +
t
1−t
¶
¸
¤
£
+ Π = (1 − γ) (1 + L0 )M + P Y (17)
where the second equality follows from the fact that the last two terms inside the brackets of the
middle expression are equal to total nominal income, P Y. Money market equilibrium requires
that total money supply, given by Mas = (1 + L0 )M, is equal to total money demand or
£
¤
(1 + L0 )M = (1 − γ) (1 + L0 )M + P Y
14
(18)
Rearranging, total output, Y, can be expressed as a function of total real money balances.
Y =
M
γ
(1 + L0 ) .
1−γ
P
(19)
Equation (19) implies that total demand for variety j can be expressed as the following function
of real money balances (details appear at the end of Appendix 9.1).
Cja
3.3
=
µ
Pj
P
¶−θ
γ(1 + L0 )
1−γ
µ
M
P
¶
(20)
.
Price setting by firm j and its derived demand for labor
The real profits of firm j are given by equation (6). Using the production function, (7), to obtain
the derived demand for labor, Ldij , the profit function may be expressed as
1
Πij
Wi
Pj
= Cja −
(Cja ) α ,
P
P
P (1 − t)
j ∈ [0, 1]
(21)
where demand for variety j (Cja ) is given by (20). Maximization of this expression subject to
(20) and a given value of P , yields the following equilibrium relative price for the good of firm j
Pj
0
= Ψ1
P
µ
θ
θ−1
0
where D ≡ α + θ(1 − α) > 0 and Ψ1 ≡
¶ Dα µ
Wi
P (1 − t)
¶ Dα µ
¡ 1 ¢ Dα ³ γ(1+L0 ) ´ 1−α
D
α
1−γ
M
P
¶ 1−α
D
(22)
> 0. Inspection reveals that the
equilibrium relative price is an increasing function of the gross real wage, of the level of real
money balances in the economy and a decreasing function of the elasticity of substitution, θ.16
16
³
θ
θ−1
α
´D
is a decreasing function of θ both because
θ
θ−1
is an increasing function of θ.
15
is a decreasing function of θ, as well as because D
The demand for labor by firm j can be obtained by using the production function in (7)
to express the demand for labor in terms of Cja and by substituting out
Pj
P
from the expression
for Cja by using the profit maximizing price in (22). This yields
Ldij
= Ψ2
µ
Wi
P (1 − t)
¶− Dθ µ
M
P
¶ D1
(23)
where Ψ2 is a combination of parameters whose explicit form appears in the appendix. Thus,
labor demanded by the firm is a decreasing function of the gross real wage and an increasing
function of real money balances.
3.4
Choice of money supply by the central bank
It is shown in Appendix 9.2 that approximate expressions for the inflation rate and the rate of
unemployment are given by the following functions of the money supply and of nominal wages
D
log
π = [D log Ψ1 ] + (1 − α)m + ατ +
1−θ
µZ
0
1
α(1−θ)
D
Wj
dj
¶
− p−1
³ ´
³ ´
c2
c1 − log W
u = [l0 − log Ψ2 + (1 − θ) log Ψ1 ] − m + τ + log W
(24)
(25)
where π ≡ p − p−1 , m is the (natural) logarithm of M, Wj is the nominal wage paid by firm j,
α(1−θ)
−θ
R
R
c2 ≡ 1 W D dj and Ψ1 is defined in the appendix. Note that an increase
c1 ≡ 1 W D dj, W
W
j
j
0
0
in the tax wedge, τ , raises both inflation and unemployment.
The CB chooses the (logarithm of the) money supply m to minimize the objective function
(8), where I ∈ [0, ∞) denotes the degree of CBC, subject to (24) and (25). The first-order
16
condition of the CB’s problem,
∂Γ
∂m
= 0, yields the optimal reaction function:
¸
1 − α(1 − α)I
·τ +
m = Ψm +
K
∙
¸
³ ´ ∙1¸
³ ´
(1 − θ) − D(1 − α)I
c1 −
c2
+
· ln W
· ln W
(1 − θ)K
K
∙
(26)
where K ≡ 1 + (1 − α)2 I > 0, and Ψm is a constant whose explicit form appears in Appendix
9.2.
Equation (26) is the CB reaction function. It implies that the sign of the response of the
money supply to an increase in the tax wedge, τ , depends on the degree of CBC, I. Depending
on whether CBC is larger or smaller than
1
α(1−α)
the CB reacts to an increase in the tax wedge by
reducing or raising the money supply. The intuitive reason is that an increase in the tax wedge
raises both inflation and unemployment. Although the CB dislikes those changes it cannot fully
offset both since it has only one instrument. If it is sufficiently conservative it partially offsets
the increase in inflation in spite of the fact that this aggravates unemployment. If it is sufficiently
liberal it partially offsets the increase in unemployment in spite of the fact that this aggravates
inflation.
To illustrate the reaction of the money supply rule in (26) to nominal wages consider the
case in which all nominal wages in the economy increase by the factor ξ > 1. Staightforward
calculations then show that the elasticity of the money supply with respect to ξ (defined as
dm
)
dξ/ξ
is equal to
1−α(1−α)I
.
1+(1−α)I 2
Thus, an increase in nominal wages induces a contraction in the
money supply if and only if the CB is sufficiently conservative or, more formally, if and only
if I >
1
.
α(1−α)
This corroborates the robustness of a related result in Coricelli, Cukierman and
Dalmazzo (2006), in a framework that abstracts from the impact of fiscal policy on monetary
policy and the economy.
17
3.5
Union i’s choice of nominal wage and symmetric equilibrium.
Each monopolistic union i, i ∈ {1, 2, .., n}, sets the same nominal wage Wi for all its members
so as to maximize the typical member’s expected utility, (9). As a consequence, all firms whose
workforce is controlled by union i pay the same nominal wage. In setting its wage the union
takes the demand function facing its workforce and the nominal wages of other unions as given
and calculates the impact of this choice on the real wage and the unemployment rate among
its members, and through them, on the expected utility of a typical member. In calculating
the impact of its choice the union’s management takes into consideration the direct impact of
its action, as well as the impact through the subsequent monetary policy of the CB. The latter
involves the monetary policy reaction of the CB in equation (26) and its consequences for the
wage and unemployment rates among union members.
Appendix 9.3 provides the technical details of the solution to the choice of nominal wage
by each union. First, we let union i maximize (9) with respect to Wi , taking as given the wages
set by other unions. Second, since all unions are identical, we focus on a symmetric solution in
which Wi = W , i = 0, 1..., n. Third, by using the approximation log(1 − x) ∼
= −x, we express
the symmetric solution to the unions problems in terms of a wage-premium, φ. This premium
is defined as the percentage (positive) deviation of the equilibrium real wage induced by the
unions actions from the competitive wage level.17 Formally
g
φ ≡ (w + τ − p) − (wc + τ − p) ≡ wrg − wrc
17
(27)
The discussion in the text implicitly focuses on the case in which the underlying parameters are such that
the wage premium is positive. In view of the union’s objective function in (9), a negative wage premium cannot
arise in equilibrium. The reason is that, in the range of negative premia, there is an excess demand for labor.
Hence, by raising the premium at least till zero the union can raise the real wage of its member without increasing
unemployment. In general there may exist combinations of parameters at which there will be a corner solution
at a zero premium.
18
g
where wrg and wrc
denote respectively the (logarithms of the) gross equilibrium real wage rate in
the presence of unions and its counterpart under a competitive labor market. The competitive
gross real wage is defined as the real wage at which the labor market clears so that the rate of
unemployment is zero. Appendix 9.3 shows that it is given by18
g
wrc
¸
∙
1
θ−1
.
= log α
θ (L0 )1−α
(28)
Appendix 9.3 also shows that the wage premium, φ, is determined implicitly by the following
equation:
1
f (φ) = aφ + b (1 − φ) 1−α + c = 0
(29)
where
¸
¶
∙ µ
1−β
R · Zu
+
a ≡ − Zw
g
1−α
(1 − t)ewrc
¶
µ
1 − γ α (1 − α) I J
b ≡ −
γ
nK
(1 − t)
R · Zu
c ≡ Zw +
g − (1 − β)Zu = 0.
(1 − t)ewrc
Here J is a positive combination of parameters defined in Appendix 9.3 and Zw and Zu are given
by:
1
Zw ≡ 1 −
> 0;
n [1 + (1 − α)2 I]
∙
¸
1
θ(n − 1)
(1 − α)I
Zu ≡
+
> 0.
n α + θ(1 − α) 1 + (1 − α)2 I
(30)
Zw measures the overall elasticity of the union’s net real wage with respect to a change in its
nominal wage, and Zu measures the overall elasticity of the union’s unemployment rate with
18
g
wrc
is an increasing function of the contribution of labor to production as characterized by α and a decreasing
function of the relative supply of labor, L0 and of the degree of competitiveness on goods markets as measured
by the elasticity of substitution, θ,across varieties.
19
respect to a change in the union’s nominal wage.19
Approximating the function f (φ) in (29) linearly around a zero wage premium and rearranging, we obtain the following explicit solution for φ:20
φ∼
=
h
Zw
Zu
+
h³
i ³ ´
α(1−α)I ¡ J ¢
− (1 − β) − 1−γ
γ
nKZu
1−t
i ³ ´
¡
¢ .
1−γ
R
αI
J
−
+ (1−t) exp(w
g
γ
nKZu 1−t
rc )
R
g
(1−t) exp(wrc
)
1−β
1−α
´
Zw
Zu
(31)
At identical quantities the utility from a unit of aggregate consumption is likely to be substantially larger than the utility from a unit of real money balances. This means that γ is likely to
be close to one, implying that the ratio
value of
1−γ
γ
1−γ
γ
is likely to be relatively small. At a sufficiently small
the sign of the expression for the wage premium in (31) is determined by the signs
of the first expression in the numerator and the first expression in the denominator. The first
expression in the denominator is unambigously positive while the first expression in the numerator may be of either sign in general. A sufficient, but not necessary, condition for a positive
¡ ¢
g
wage premium is ZZwu + WR > 1 where W
= (1 − t) exp(wrc
) is the real net competitive wage
P c
( P )c
rate.21
Finally, the equilibrium values of the economy-wide unemployment rate, u, and of the
inflation rate, π, can be expressed as linear functions of the wage-premium φ. Appendix 9.3
shows that their equilibrium values are given by:
u=
φ
1−α
19
(32)
Those elasticities also play a crucial role in the solution to the union’s problem in Coricelli, Cukierman and
Dalmazzo (2006). Further details on their meaning and role appear in that paper.
20
The details appear in appendix 9.3.
21
For ballpark reasonable values of the parameters such as θ = 2, n = 5, α = 0.9 and I = 5, ZZwu ∼
= 0.6 and this
condition reduces to WR > 0.4. This implies that, for those parameters, a sufficient condition for a positive
( P )c
wage premium is that the ratio between the value of leisure and the competitive real wage exceed 40%.
20
and
π=
φ
.
(1 − α)2 I
(33)
Thus, inflation, unemployment, and therefore employment and the total product, depend on the
wage premium. It is shown in the next subsection that the share of labor in total income also
depends on the premium. Establishing the effects of economic structure, policy variables and
institutions on the premium is therefore an important intermediate step in finding their long
run effects on the economy. Section 5 below reports the effects of various structural parameters
on the wage premium.
4
The functional distribution of income, real marginal
costs and the markup
4.1
Labor share in the presence of collective bargaining
A central objective of labor unions is to push the real wage of their members above the competitive level. Hence, imperfect competition in the labor market generally affects the share of
labor in total income. But this does not necessarily imply that unions actions also raise the
share of labor in national income. Whether this is the case or not depends on the magnitudes
of the elasticity of the demand for labor by employers and on the elasticity of total income with
respect to the real wage. This subsection derives the implications of our framework for the share
of labor in total income in the presence of collective bargaining by unions.
As a benchmark we start by deriving the share of labor under monopolistically competitive product markets when the labor market is competitive. In a perfectly competitive labor
market the rate of unemployment in the model is zero and the gross real wage is given by
21
equation (28).22 Hence total employment is L0 and the (gross) share of labor is
ScL ≡
g
)
L0 exp(wrc
g
= L1−α
exp (wrc
).
0
α
L0
(34)
In the presence of collective bargaining the wage premium is normally positive so that
the rate of unemployment is positive as well. In this case labor share in a symmetric general
equilibrium is
SuL
≡
W
(1
P (1−t)
− u)L0
[(1 − u)L0 ]α
=
L1−α
0
µ
¶1−α
W
φ
1−
P (1 − t)
1−α
(35)
where the last equality follows by using (32). The definition of the wage premium in (27).implies
that
W
g
= exp(wrc
) exp(φ).
P (1 − t)
(36)
Inserting this expression in (35), using (34) and rearranging
SuL
≡
W
(1
P (1−t)
− u)L0
Lα0
=
ScL exp(φ)
µ
1−
φ
1−α
¶1−α
≡ ScL g(φ).
(37)
Note that, if g(φ) is smaller than one, the share of labor under monopolistic competition and
collective bargaining in the labor market is smaller than the share of labor, ScL , under monopolistic competition and a competitive labor market. As established in the following proposition
g(φ) is indeed smaller than or equal to one.
Proposition 1:
(i) g 0 (φ) < 0.
(ii) g(φ) ≤ 1 for all positive values of φ with the equality holding when the wage premium
is zero.
22
Obviously, this statement abstracts from frictional and search unemployment of the type stressed by
Mortensen and Pissarides (1999). More generally one could think of the "zero unemployment" restriction in
our model as a benchmark that sets frictional unemployment to zero by means of a normalization.
22
Proof:
(i) g0 (φ) = − φ exp(φ)
1−α
1
α < 0.
φ
(1− 1−α
)
(ii) g(0) = 1. Since, from part (i), g(φ) is decreasing in φ, it is, a fortiori, smaller than 1
at all positive values of φ. QED
The proposition implies that SuL is generally smaller than ScL and that the difference between
those two shares increases with the wage premium, φ.23 Thus, when the wage premium is
positive, imperfect competition in the labor market reduces the share of labor below its share
under a competitive labor market.
4.2
A Remark on real marginal costs and the markup
Real marginal cost and the markup are concepts that play a central role in New Keynesian
theories of inflation and business fluctuations.24 Invariably, those models assume that the labor
market is competitive.25 It is therefore interesting to examine how those concepts are altered, if
at all, when it is recognized that labor markets are non competitive. Appendix 9.4 shows that
the equilibrium values of real marginal costs and of the markup, µ, are given respectively by
MCr =
and
µ≡
Pj
P
MCr
=
θ−1
θ
1
θ
.
=
MCr
θ−1
(38)
(39)
The second result confirms for this model that, with flexible prices, firms always set prices so as
to attain the profit maximizing markup, which for Dixit-Stiglitz utility is a constant given by
23
One reason for this result (as can be checked from (23)) is that the elasticity of labor demand with respect
to the gross real wage is larger than one.
24
Well known examples are Clarida, Gali and Gertler (1999) and Woodford (2003).
25
A recent exception is Gnocchi (2005).
23
equation (39).
5
General equilibrium comparative statics
This section summarizes comparative static results effects of economic structure, fiscal parameters and CBC on the economy. The propositions highlight the effects of various structural
parameters on the wage premium and through it on related variables like unemployment and
inflation. In deriving the long run impact of a given structural or fiscal policy variable on the
economy the relations between the wage premium on one hand, and unemployment and inflation
on the other given by equations (32) and (33) are utilized. Proofs appear in Appendix 9.4.
5.1
Impacts of goods’ market competitiveness and of centralization
of wage bargaining
Proposition 2:
(i) A higher elasticity of substitution, θ, is associated with a higher competitive real wage
rate.
(ii) Provided 1 − γ is sufficiently small, α > β ≥ 0, and the participation constraint in
(10) is satisfied a higher θ is associated with a lower wage premium, higher employment and
lower inflation.
The condition α > β ≥ 0 states that the exponent of labor in the production function
is larger than the replacement ratio. Since the exponent of labor is at least two thirds and
replacement ratios in most developed economies are smaller than a half, this condition is likely
to be satisfied in practice.
Proposition 3: Provided 1 − γ is sufficiently small, α > β ≥ 0 and the participation
constraint in (10) is satisfied, more centralization of wage bargaining,
24
1
,
n
is associated with a
lower wage premium, higher employment and lower inflation.
Propositions 2 and 3 highlight a striking difference between the impacts of competitiveness in the goods’ and in the labor market on the economy. Whereas a higher degree of
competitiveness in the goods’ markets is associated with a lower wage premium, higher employment and lower inflation, more decentralization of wage bargaining in the labor market is
associated with a higher wage premium, lower employment and a higher rate of inflation. The
origin of those opposite effects of competitiveness is that, since they are large wage setters,
unions partially internalize the impact of their wage decisions on the economy, whereas firms do
not.
5.2
The impact of fiscal parameters
The propositions in this subsection focus on the impact of fiscal policy instruments on the wage
premium, employment and inflation.
Proposition 4: Provided 1 − γ is sufficiently small and α > β ≥ 0, a higher tax wedge,
t, is associated with a higher wage premium, lower employment and higher inflation.
Proposition 5: Provided 1 − γ is sufficiently small, higher unemployment benefits, as
represented by a higher replacement ratio, β, are associated with a higher wage premium, lower
employment and higher inflation.
Thus, as expected, a higher replacement ratio, by raising the bargaining power of unions,
raises the wage premium and unemployment. Provided α > β ≥ 0 a higher tax wedge has a
similar impact.26 Those results confirm, within a framework that also features explicit monetary
policy, similar results found in Alesina and Perotti (1997) . At the same time, by deriving a
sufficient condition for a positive association between the wage premium and the tax wedge,
26
Table 1 in Ardagna (2007) suggests that the replacement ratio in European countries is about 0.25 and that
it varies between a minimum of 0.17 and a mximum of 0.32.
25
proposition 4 qualifies their result.
As can be checked from equation (23), the demand for labor is elastic implying that,
although the real wage increases when the replacement ratio increases, the share of labor goes
down. This implies that higher replacement ratios are associated with a lower wage bill so that
total labor income may actually go down. A related result appears in Ardagna (2007). In a
framework with both public and private employment and unions she finds that fiscal policies
designed to raise the disposable income of one group of workers can be more than compensated
by their impact on employment.
5.3
The impact of central bank conservativeness (CBC)
Proposition 6: Provided 1 − γ is sufficiently small and the participation constraint in (10) is
satisfied, a more conservative central bank (a higher I) is associated with a lower wage premium,
higher employment and lower inflation.
This result confirms and qualifies, within a fully microfounded framework that features
both fiscal and monetary policies, a similar result found in Soskice and Iversen (2000) and
Coricelli, Cukierman and Dalmazzo (2006).
5.4
Cross effects among CBC, CWB and the tax wedge
Propositions 3 and 6 above show that higher centralization of wage bargaining (CWB) and
higher central bank conservativeness (CBC) exert moderating effects on the wage premium and
lead, through this mechanism, to higher employment. The following proposition shows that in
the presence of a higher tax wedge, t, those moderating effects are weaker.
Proposition 7: Provided 1 − γ is sufficiently small, α > β ≥ 0 and the participation
constraint in (10) is satisfied,
(i)
dφ
dI
is smaller in absolute value when the tax wedge, t, is higher,
26
(ii)
6
dφ
1
d( n
)
is smaller in absolute value when the tax wedge, t, is higher.
Welfare analysis.
This section evaluates the effects of competitiveness in the goods and labor markets, of fiscal
policy parameters and of CBC on aggregate welfare. We use a Benthamite measure of social
welfare that is based on the sum of the indirect utility functions, vba , of all individuals in the
general equilibrium. Total welfare equals the welfare of employed workers plus the welfare of
unemployed workers, each weigthed by its appropriate proportion in the labor force, plus the
welfare of employers. Aggregate welfare is given, therefore, by
vba = ((1 − u)vEW + uvUW ) L0 + vE .
(40)
Substituting (15) into (40) and rearranging
M Bu + χ
W
Π
+
L0 + (1 − u) L0 + RuL0 +
P
P
P
½ P
¾
M
Π
tW
W
= (1 + L0 ) + (1 − u)
+ (1 − u)
L0 + + uRL0
P
(1 − t)P
P
P
vba = (1 + L0 )
(41)
where the last equality follows from the government budget constraint in (11) and the reader is
reminded that χ ≡ T RW +
1
T RE .
Lo
Gross real income is equal to the sum of (real) taxes, net
wages and profits. Hence
½
Y = (1 − u)
W
tW
+ (1 − u)
(1 − t)P
P
27
¾
L0 +
Π
.
P
(42)
Using (42) in (41), average welfare per individual can be expressed as
vb ≡
M Y + RuL0
vba
+
=
.
1 + L0
P
1 + L0
(43)
It is shown in Appendix 9.6 that vb can be expressed as the following function of the wage
premium:
vb(φ) = Ψ5 · [1 − φ]
α
1−α
∙
µ
¸α
¶
φ
Lα0
L0
φ
1−
·R
+
+
1 + L0
1−α
1 + L0 1 − α
(44)
where Ψ5 > 0 is defined in equation (102) in the appendix. Inspection of (44) suggests that
except for the elasticity of substitution, θ, the main parameters of interest affect welfare only
through the wage premium, φ. Consequently an important intermediate step in determining the
effects of those parameters on welfare involves finding the effect of the wage premium on vb(φ).
An increase in the wage-premium, φ, has three effects on vb(φ): (i) it reduces utility by reducing
income from production, (ii) it increases utility by raising the number of unemployed, so that
their total leisure increases and (iii) it reduces utility by reducing real balances.
Based on (44) we can find how goods’ market competitiveness, CWB, the tax wedge,
the replacement ratio and CBC, affect welfare through φ by totally differentiating vb(φ) with
¡
¢
respect to θ, n1 , t, β, I . The resulting expressions can then be combined with propositions 2-6
to establish the effect of each of those parameters on welfare. The results are summarized in
the following propositions and demonstrated in Appendix 9.6:
Proposition 8: For (1 − γ) sufficiently small,
(i) The higher the replacement ratio, β, the lower social welfare.
If in addition, the participation constraint and the conditions α > β ≥ 0 and
α [(1 − u)L0 ]α−1 > R,
28
(45)
are satisfied, then:
(ii) The higher CWB, n1 , the higher social welfare,
(iii) The higher the tax wedge, t, the lower social welfare.
(iv) The higher CBC, I, the higher social welfare.
Condition (45) requires that the marginal contribution to output (and therefore consumption)
from an additional employed individual has to be greater than the value of leisure foregone by
the individual. It is a necessary condition for some positive employment to be socially desirable.
The intuition underlying the results in proposition 8 follows. A higher degree of centralization of wage bargaining, by raising the degree of internalization of the macroeconomic
impact of its actions, moderates the wage demands of each union, leading to lower real wages,
more employment, higher income and higher welfare. By raising the wage demands of unions
increases in the replacement ratio and the tax wedge lead to higher real wage costs to firms,
higher unemployment, lower aggregate income and lower welfare. Finally, a higher degree of
CBC, by sending a signal to unions that the CB will react to higher wage demands with a
stronger contraction of aggregate demand, moderates their wage demands raising employment,
income and welfare.
7
Fiscal policy and special interests.
To this point, fiscal policy has been taken as exogenous. This section extends the analysis by
adding a first stage during which a political authority picks fiscal instruments, anticipating the
reactions of labor unions, the central bank and price setters in subsequent stages. Formally, the
fiscal authority is modeled as a Stackelberg leader.
It is well known that the motives of political authorities and of social planners are not
fully aligned. Although this does not necessarily mean that politicians do not care at all about
social welfare, it usually implies that they also care about redistribution in favor of particu29
lar constituencies. We therefore endow fiscal authorities with an objective function that is a
weighted average of social welfare and of redistribution in favor of particular constituencies. In
particular, government’s objective function is given by:
Υ=δ·
hχ
P
i
L0 + (1 − δ) · vb
As the discussion preceding equation (11) clarifies the term
χ
L
P 0
(46)
on the right-hand side of (46)
represents the total amount of real transfer payments in favour of preferred constituencies.27
Thus, the parameter δ ∈ [0, 1] represents the weight given by government to redistribution in
favor of "special interests" and 1− δ represents the weight given to, vb - - the indirect utility of
the average individual in the economy that is given by (44). For simplicity we abstract from
public goods and unemployment benefits by assuming that fiscal policy instruments consist only
of taxation and of lump sum redistribution so that B = β = 0. As a consequence government’s
budget constraint in (11) specializes to
χ
W t(1 − u)L0
L0 =
≡ T (t).
P
P
1−t
(47)
The left hand side of (47) represents total real redistribution and the right hand side represents
total real taxes.
The instrument of fiscal policy - - the tax rate t - - is chosen to maximize (46) subject
to the budget constraint in (47). To develop some intuition about the mechanisms underlying
government’s choice it is convenient to start with two extreme particular cases: (i) a government
that has no interest in redistribution (δ = 0) and, (ii) a government that only cares about
redistribution in favour of supporters (δ = 1).
Since χ ≡ T RW + L10 T RE is the average transfer per worker in the economy χL0 represent total nominal
transfers. Note that, although transfers are measured per worker, they can accomodate any pattern of transfers
between workers and entrepeneurs.
27
30
Case (i). A government that only cares about social welfare gives no weight to redistribution (δ = 0) and sets t so as to maximize vb. Part (iii) of proposition 8 implies that welfare
is maximized when t = 0. Hence, like a Benthamite social planner, a government with no
redistributional concerns does not impose taxes.28
Case (ii) A totally partisan government that only cares about the special interests of its
preferred constituency (δ = 1) chooses the tax wedge, t, so as to maximize T , the amount of
funds available for redistribution in (47). Formally such a government sets the tax wedge, t, so
that (for an internal solution) the condition
dT (t)
dt
dT (t)
dt
= 0 is satisfied.29 In general, the derivative
may be of either sign, since a higher tax rate reduces economic activity. However, a rational
government will never operate on the inefficient side of the Laffer-curve. That is, the equilibrium
value of t must be such that
7.1
dT (t)
dt
≥ 0. Further details appear at the end of Appendix 9.6.
Characterization of t and of the size of government in the general
case
In order to characterize the equilibrium values of t and of total redistribution in this intermediate
case, we need to express all the components of T in (47) as a function of t. Since both the real
wage and unemployment depend on t via the wage premium,we start by expressing T in terms
of the wage premium, φ(t), where the notation highlights the dependence of the wage premium
on the tax wedge. Equation (75) in Appendix 9.3 implies
W
=
P
µ
W
P
¶
c
1
(1 − t)
g
∼
)
.
= exp(wrc
(1 − φ(t))
(1 − φ(t))
28
(48)
Obviously, this extreme conclusion is a consequence of the implicit assumption that utility from public goods
is zero. In the presence of utilty from public goods there will be, in this case, some taxation but only to finance
the public good.
29
As pointed out by Meltzer and Richard (1981) such a political equilibrium arises when the median voter in
their model does not work.
31
Inserting (32) and (48) into the middle expression in (47), rearranging and inserting the resulting
expression into (46) the objective function of the fiscal authority can be expressed as the following
function of t
g
)
Υ(t) = δ · T + (1 − δ) · vb ∼
= δ exp(wrc
∙
¸
t · L0
φ(t)
1−
+ (1 − δ)b
v(t).
(1 − φ(t))
1−α
(49)
For an internal solution the tax rate, t∗ , that maximizes government’s objectives in (49) has to
satisfy the first-order condition
dΥ(t∗ )
dT (t∗ )
db
v (φ(t∗ )) dφ
=δ
+ (1 − δ)
=0
dt
dt
dφ
dt
(50)
where the functions T (t) and vb(t) are given by (47) and (44) respectively. Part (iii) of proposition
8 implies that
de
v (φ(t∗ ))
dφ
dφ
dt
dT (t∗ )
dt
< 0 and, from proposition 4,
right hand side of (50) is negative. It follows that
> 0. Hence the second term on the
> 0. This confirms that government
operates on the efficient side of the Laffer curve also in the intermediate case. It implies that
the equilibrium size of redistribution (and therefore of government) is an increasing function of
the tax wedge.
7.2
Comparative statics, CBC and the size of government
Application of the implicit function theorem to (50) yields
dt∗
=−
dδ
dT (t∗ )
dt
de
v (φ(t∗ )) dφ
dφ
dt
SOC(t∗ )
−
where SOC(t∗ ) < 0 is the second order condition for government’s decision problem. Proposition
4 and part (iii) of proposition 8 imply that
dT (t∗ )
dt
proposition.
32
−
de
v(φ(t∗ )) dφ
dφ
dt
> 0. This leads to the following
Proposition 9: Governments with higher relative distributional concerns (higher δ 0 s)
set higher tax wedges and are larger.
The impact of CBC on the tax policy of Government is generally ambiguous since an
increase in CBC triggers two opposing effects on the marginal impact of the tax wedge on total
tax collections and redistribution. On one hand, by raising
dT
dt
for a given wage premium, an
increase in I tends to increase the revenue enhancing marginal impact of t on tax collections.
This effect encourages Government to raise the tax wedge. On the other hand, higher CBC,
by magnifying the adverse effect of t on the wage premium and unemployment operates in
the opposite direction.30 The following proposition (demonstrated in Appendix 9.6) provides a
sufficient condition for the dominance of the first effect.
Proposition 10: Provided the cross-derivative,
d2 φ
,
dt·dI
is not too large, the tax rate, t∗ , set by
fiscal autorities is increasing in the level of CBC, I.
7.3
An implication for the social desirability of strict inflation targeting
An implication of proposition 10 is that, when the effect of CBC on the (now endogenously
determined) tax wedge is taken into consideration, the optimal long run level of conservativeness
may no longer be infinite. The reason is that although, given the tax wedge, an ultra conservative
CB is optimal (by part (iv) of proposition 8) it may induce government to raise the tax wedge
which (by part (iii) of proposition 8) reduces welfare. The upshot is that, when the impact
of conservativeness on welfare through the choice of tax wedge by government is taken into
consideration, an ultra conservative CB, or equivalently - - a strict inflation targeter - - may no
longer be optimal. An analytical formulation of this point appears in Appendix 9.7
30
A higher I also triggers two opposing effects on the marginal impact of t on welfare.
33
8
Concluding remarks
This paper explores the implications of interactions between fiscal policy and monetary policymaking institutions in an economy with imperfectly competitive labor and goods markets with
sticky wages and flexible prices. Such a framework appears to be a more realistic description
of economic realities in Europe than some New Keynesian models featuring a competitive labor
market with sticky prices and flexible wages for at least two reasons. First, and most obviously,
the labor force in Europe is largely unionized. Second, recent evidence from the ECB inflation
network supports the view that wages are stickier than prices.
As emphasized by Soskice and Iversen (2000) and Coricelli, Cukierman and Dalmazzo
(2006) and others, one basic difference between a competitive labor market and a labor market
characterized by large wage setters is that in the latter case, due to the fact that unions internalize the response of the central bank (CB) to their wage setting decisions, the level of CBC
affects unemployment and other real variables even in the long run. A basic consequence of this
"strategic effect" is that, in the absence of shocks (and therefore no benefit from stabilization
policy) strict inflation targeting improves performance not only on the inflation front but it also
reduces unemployment and is therefore socially optimal.
The result above is obtained within frameworks that abstract from fiscal policy. This
paper discusses the impact of fiscal policy decisions on this mechanism, as well as on the reverse
causality from CBC to the choice of fiscal policy variables like labor taxes and redistribution.
Two results stand out in this context. First, as long as labor taxes and unemployment benefits are given exogenously the social desirability of strict inflation targeting is robust to the
introduction of fiscal policy. However, higher conservativeness on the part of the CB often induces governments with redistributional concerns to raise taxes, and this reduces social welfare.
As a consequence flexible inflation targeting may be socially optimal even in the absence of
34
stabilization policy.31
We use the framework of the paper to investigate the impact of economic structure, fiscal
policy, central bank conservativeness (CBC), and their interactions on economic performance in
the long run. A central determinant of economic performance in our model is the wage premium
- - defined as the percentage (upward) deviation of the equilibrium real wage in the presence of
unions from its counterpart when the labor market is competitive. The paper investigates the
long run impact of labor taxes, the replacement ratio, goods market competitiveness, centralization of wage bargaining and CBC on the wage premium, and through it, on unemployment,
inflation and related macroeconomic variables. Under reasonable restrictions higher values of the
tax wedge and of the replacement ratio are associated with a higher wage premium, while higher
values of centralization of wage bargaining and of CBC are associated with a lower premium.
Social welfare, defined as the sum of all individual utilities, is negatively related to the
wage premium. As a consequence higher values of the tax wedge and of the replacement ratio
are associated with lower welfare. By contrast higher values of good market competitiveness, of
centralization of wage bargaining and of CBC are associated with higher welfare.
Blanchard and Giavazzi (2003) note that since the end of the seventies there has been
a persistent decline in labor share and a persistent increase in the rate of unemployment in
Germany, France, Italy and Spain. Our model predicts that increases in labor taxes should
cause both of those developments. It is therefore interesting to examine what happened to those
taxes in Europe since the end of the seventies. Table 1 in Ardagna (2007) suggests that (for a
sample of ten European countries that includes the four countries above) the average effective
tax rate on labor income has risen from about 32% in the mid seventies to almost 43% during
the first half of the nineties.32
31
By contrast in Rogoff (1985) classic framework, since CBC does not affect employment in the long run, a
strict inflation targeter is socially optimal in the absence of stabilization policy.
32
Blanchard and Giavazzi (2003) explore an alternative hypothesis that relies on changes in the bargaining
power of labor.
35
Finally, one intriguing consequence of our framework is that the share of labor in the
presence of unions is lower than this share when the labor market is competitive. Furthermore,
this share is a decreasing function of the wage premium.
9
9.1
Appendix.
The individual consumer’s problem.
Omitting, without loss of generality, the class index, c, and using the first-order conditions with
respect to varieties j and s, we obtain
Cj
=
Cs
µ
Pj
Ps
¶−θ
,
(51)
for any (j, s).
Solving for Cj from (51), substituting it into (4) and using (3), we obtain
Cj =
µ
Pj
P
¶−θ µ
A−M
P
¶
(52)
.
¡ ¢
Raising both sides of (52) to power θ−1
, integrating over j and raising the result to power
θ
¡ θ ¢
. Hence
, one also obtains that C = A−M
θ−1
P
Cj =
µ
Pj
P
¶−θ
(53)
C.
Combining the first-order conditions with respect to variety j and money, M, we obtain
(1−γ)Pj
.
M
−1
θ
γCj
C
θ−1
θ
=
Using (53) to substitute Cj out from this expression and rearranging
γ
C=
1−γ
36
µ
M
P
¶
.
(54)
Each individual’s demand for variety j, given by (51), can therefore be rewritten as
Cj =
µ
Pj
P
¶−θ µ
γ
1−γ
¶µ
M
P
¶
.
(55)
The demand for nominal money (equation (13) in the text) is obtained by substituting this
expression for Cj in the individual’s budget constraint (4) and by rearranging. Substituting
(13) into (54), the consumption aggregator, C, can be expressed as
C=γ
A
.
P
(56)
Similarly, using (13), we can rewrite (55) as
Cj = γ
µ
Pj
P
¶−θ µ ¶
A
.
P
(57)
As demonstrated later, in a symmetric equilibrium individual behavior differs only among three
classes of individuals: employed workers, unemployed workers and employers. Aggregating (57)
over the mass of individuals in the economy and using equations (4) and (5), total demand for
variety j is given by
Cja
=γ
µ
Pj
P
¶−θ
Aa
=γ
P
µ
Pj
P
¶−θ ∙
¸
M
(1 + L0 ) + Y .
P
(58)
The expression for total demand for variety j in equation (20) of the text is obtained by inserting
(19) into (58) above and by rearranging. Finally, the indirect utility function in equation (14)
in the text is obtained by substituting (13) and (56) into the utility function (1).
37
9.2
Derivation of expressions for inflation, unemployment and the
Central Bank’s reaction function
To derive the reaction function of the CB we have to express inflation and unemployment in
terms of the money supply. To obtain an expression for inflation we raise the equilibrium relative
price of each firm in (22) to the power of (1 − θ) and integrate over firms. Using (3) and taking
natural logarithms of the resulting expression we obtain:
1
1−α
α
0 = log Ψ1 +
(m − p) + (τ − p) +
log
D
D
1−θ
µZ
1
α(1−θ)
D
Wj
dj
0
¶
(59)
where τ ≡ − log(1 − t), and m and p denote the natural logarithms of M and P respectively,
¸ Dα
∙
³
´ 1−α
α
γ(1+L
)
θ
0
. The price-level, p, is obtained by rearranging (59) and by
and Ψ1 ≡ (θ−1)α
1−γ
substituting it into the definition of inflation.
To obtain the rate of unemployment, u, we aggregate (23) over firms and take logs.
Defining log(L0 ) ≡ l0 , we obtain:
1
θ
u ≡ l0 − l = [l0 − log Ψ2 ] − (m − p) − (p − τ ) − log
D
D
d
where Ψ2 ≡
³
θ
(θ−1)α
´− Dθ ³
γ(1+L0 )
1−γ
´ D1
µZ
0
1
−θ
D
Wj dj
¶
(60)
. Equation (25) in the text is obtained by using (24) to
substitute out p = π + p−1 in (60). Substituting (24) and (25) into (8), the Central Bank’s
objective function can be rewritten as:
³ ´o2
n
³ ´
c2
c
Γ = [l0 − log Ψ2 + (1 − θ) log Ψ1 ] − m + τ + log W1 − log W
+
¾
½
2
³ ´
D
c
+I · [D log Ψ1 ] + (1 − α)m + ατ +
log W1 − p−1
1−θ
(61)
The reaction function in equation (26) in the text is obtained by maximizing (61) with respect
38
to m. The constant, Ψm , is given by
Ψm ≡
l0 − log Ψ2 + [(1 − θ) − (1 − α)ID] log Ψ2 + (1 − α)Ip−1
1 + (1 − α)2 I
Derivation of Union i’s choice of the nominal wage, Wi .
9.3
Union i chooses Wi so as to maximize a member’s expected utility (9). ui is equal to
L0
1
− log Ldi = l0 + log − lid =
n
n
θ
1
= [l0 − log Ψ2 ] + [wi − p + τ ] − (m − p)
D
D
(62)
ui = log
where the second equality follows from (23) and the fact that union i controls
R ni
i−1
n
i
n
i−1
n
L0 dj =
R
workers, and the number of union i members demanded by firms is Ldi =
Ldij dj =
³
´− Dθ ³ ´ D1
Wi
1
M
Ψ
. Differentiating (9) with respect to Wi , we obtain the union’s first-order
n 2 P (1−t)
P
L0
n
condition:
∙
1 − (1 − β)ui
P
¸ ∙
¸∙
¸
M dP
Wi dP
Wi dui
1 dM
+ R − (1 − β)
1−
+
− 2
=0
P dWi
P dWi P dWi P dWi
where
dui
1
=
dWi
DWi
½ ∙
¸ ∙
¸¾
Wi dP
Wi dM
Wi dP
θ 1−
−
−
P dWi
P dWi
M dWi
(63)
(64)
Since symmetry in wages implies symmetry in prices (i.e., Pj = P ), it follows from (22) that:
−D
M
= (Ψ1 ) 1−α
P
µ
W
P (1 − t)
39
−α
¶ 1−α
.
(65)
Imposing symmetry also on equation (62) (i.e., Wi = W for all i0 s) and using (65), it follows
that:
ui = u = Ψ3 +
where Ψ3 ≡ l0 − log Ψ2 +
log Ψ1
,
1−α
w+τ −p
, i = 1, n
1−α
(66)
and w and p are the logarithms of W and P , respectively.
Moreover the definition of the net real wage and equation (64) imply that:
1−
Wi dP
1
1
≡1−
≡ Zw
=1−
2
P dWi
n [1 + (1 − α) I]
nK
dui
1
1 θ(n − 1) + (1 − α) [n(1 − α)θ + α] I
.
=
Zu =
dWi
Wi
Wi
nKD
(67)
(68)
Finally, from the CB reaction function in (26)
1 − α(1 − α)I
dM Wi
=
.
dWi M
nK
(69)
Using (67), (68), and (69), the first-order condition (63) evaluated in a symmetric equilibrium
of the nominal wage setting game can be rewritten as:
∙
µ
∙
¸
¶¸
µ
¶
¡
¢
¢
¡
M α(1 − α)I
w+τ −p
P
1 − 1 − β Ψ3 +
R − 1 − β · Zu −
· Zw +
= 0 (70)
1−α
W
W
nK
This equation implicitly determines the real wage. But since it involves both the level
and the logarithm of the real wage it cannot be solved explicitly. For tractability reasons
it is convenient to reformulate (70) in terms of the wage-premium, φ, that is defined as the
logarithmic difference between the general equilibrium real wage in the presence of unions and
the general equilibrium real wage when the labor market is competitive. This procedure requires
40
three additional steps. First, we express
M
W
in terms of the real wage by rewriting (65) as:
−D
M/P
(Ψ1 ) 1−α
=
·
W/P
(1 − t)
µ
W
P (1 − t)
−1
¶ 1−α
(71)
.
Second, we characterize the level of the competitive real wage in terms of the model’s parameters.
Noting that, under symmetry, the rate of unemployment among union i’s members (62) is equal
to the economy-wide unemployment rate u, we can set u = 0 in (66) to determine the (logarithm
g
of the) gross competitive, wrc
. After some algebra this yields
g
wrc
¸
∙
1
θ−1
.
≡ (wc + τ − pc ) = −(1 − α)Ψ3 = log α
θ (L0 )1−α
(72)
Third, we consider the identity
∙ µ ¶¸
P
P W
1
=
W
(W/P )c W P c
where (W/P )c is the net real competitive wage. Letting x =
(73)
P
W
x∼
= 1 + (log x) and the definition of the wage premium in (27)
P
W
µ
W
P
¶
c
¡W ¢
P
c
, using the approximation
∼
= 1 + (wc − pc ) − (w − p) ≡ 1 − φ.
(74)
Substituting (74) into (73) yields
1
P ∼
1
[1 + (wc − pc ) − (w − p)] ≡
[1 − φ] .
=
W
(W/P )c
(W/P )c
(75)
Equation (29) in the text is obtained by substituting (71), (72) and (75) into (70).
Equation (31) in the text is obtained by taking a first order Taylor expansion of equation
(29) around φ = 0, setting the approximated value of f (φ) equal to zero and by rearranging.
41
The constant J in equation (31) is given by J ≡
1
1+L0
h
(θ−1)α
θ
α ³
i 1−α
1
g
exp(wrc
)
1
´ 1−α
.
Both the unemployment rate u, and the inflation rate π can be expressed as linear
functions of φ. Equation (32) in the text is obtained by noting that, in a symmetric equilibrium,
the unemployment rate among union i’s members, ui , is equal to u, and by using (66) and (72)
in (62). To obtain (33) note that the first-order condition of the CB problem (equation (8))
implies
∂u
u
∂m
∂π
+ I ∂m
π = 0, where
∂u
∂m
= −1 and
(25) and (24)). Substituting those terms into
9.4
∂π
∂m
∂u
u
∂m
= (1 − α) (this can be seen from equations
∂π
+ I ∂m
π = 0 and rearranging yields (33).
Real marginal costs and the markup in a symmetric general equilibrium
Conceptually, real marginal costs, MCr , are equal to the gross real wage,
W
,
P (1−t)
divided by the
marginal product of labor, MP so that
MCr =
W
P (1−t)
MP
=
W
P (1−t)
α(Ld )α−1
(76)
where Ld is a generic expression for the typical demand for labor. Substituting equation (23)
into (76) and rearranging
MCr =
³
W
P (1−t)
W
P (1−t)
´ Dθ
∙ ³
´− Dθ ³ ´ D1 ¸α−1 = ∙ ³ ´ D1 ¸α−1
W
M
α Ψ2 M
α Ψ2 P (1−t)
P
P
(77)
which implies that the real marginal cost is an increasing function of the real wage and of
real money balances. Increases in either of the cost of labor or of real money balances lead to
an increase in real marginal costs but the mechanisms leading to this increase differ. Given
production, an increase in the real wage raises the marginal cost directly. This effect is partially
42
offset through the fact that the firm cuts production - which raises the marginal product of
labor and reduces marginal costs. But, the extreme right hand side of (77) suggests that the
direct effect dominates so that the overall effect of an increase in the real wage is to increase
marginal costs. When real money balances (and therefore demand) increase, the firm responds
by expanding production so that, given the decreasing marginal productivity of labor, marginal
costs increase.
But this intuitive account of the economic forces at work abstracts from an additional
general equilibrium relation between real balances and the gross real wage that is summarized
by equation (65) of the appendix. That relation arises because, when the gross real wage rises,
all firms try to raise their relative prices. Since they all do that by raising their nominal prices,
none of them manages to increase its relative price. But their joint attempt to raise their
relative prices leads to an increase in the general price level which, given the nominal money
supply, depresses real money balances and with it aggregate demand. The upshot is that there
is a general equilibrium inverse relation between real money balances and the gross real wage.
Taken alone, this additional channel leads to a decrease in aggregate demand and production
so that marginal costs decrease.
It turns out that for the Blanchard-Kiyotaki-Fischer-Dixit-Stiglitz (BKFDS) framework
that underlies the structure of demand in our model (as well as in many other contemporary
models with monopolistic competition) the downward effect of a change in the real wage through
demand exactly offsets its upward effect on marginal costs through supply. This can be seen
by substituting equation (65) from the appendix into (77). After cancellation of terms and
rearrangement, this yields equation (38) in the text implying that in a symmetric general equilibrium marginal costs depend only on the elasticity of substitution, θ. Although this result is
due to the specific form of the BKFDS framework it can serve as a useful benchmark.
We turn next to the calculation of the markup in a symmetric general equilibrium. The
43
markup, µ, is defined as the real equilibrium price divided by the real marginal cost33
µ≡
Pj
P
MCr
=
1
θ
.
=
MCr
θ−1
(78)
Thus, in a symmetric general equilibrium, real marginal costs and the markup are fully determined by the degree of competitiveness in the goods markets as characterized by the elasticity
of substitution, θ. Equation (78) is a familiar expression for the markup in many other models
of monopolistic competition based on a Dixit-Stiglitz utility with competitive labor markets.
This analysis suggests that, at least for BKFDS type frameworks, the expression for the long
run markup is invariant to the structure and degree of competitiveness of labor markets.
9.5
Proofs of propositions 2-7
The derivations of the impacts on the wage premium are obtained by differentiating the expression for the wage premium in equation (31) with respect to the appropriate structural or
fiscal policy variable and by using the additional constraints specified in each proposition. The
impacts on unemployment and inflation are inferred from equations (32) and (33). In establishing the impact of a given structural or fiscal policy variable s, s = θ, n1 , t, β, I, some similar
expressions appear. Those expressions are derived once in the early stages of the proof and used
subsequently, as appropriate, to complete the proof of each separate proposition. It is convenient
to rewrite (31) as
φ∼
=
33
N1 −
D1 −
³
³
1−γ
γ
1−γ
γ
´
´
N2
(79)
D2
The second equality in (78) follows from the fact that all firms set the same price in a symmetric equilibrium.
44
where
N1
D1
µ
¶
J
Zw
R
α (1 − α) I
− (1 − β), N2 ≡
≡
+
g
Zu (1 − t) exp(wrc
)
nKZu
1−t
µ
¶
¶
µ
J
1 − β Zw
R
αI
.
≡
+
g , D2 ≡
1 − α Zu (1 − t) exp(wrc
)
nKZu 1 − t
(80)
Differentiating φ with respect to the dummy index, s, and rearranging
1−γ
dN
dD
dφ ∼ D1 ds1 − N1 ds1 + γ Q
= ³
³ ´ ´2
ds
D1 − 1−γ
D2
γ
where
Q≡
µ
1−γ
D2 − D1
γ
Under the assumption that
1−γ
γ
¶
dN2
−
ds
µ
1−γ
N2 − N1
γ
¶
dD2
dD1
dN1
+ N2
− D2
ds
ds
ds
(81)
(82)
is relatively small an approximate expression for the impact of
s on the wage premium is
1
1
− N1 dD
dφ ∼ D1 dN
ds
ds
=
ds
D12
(83)
Proof of proposition 2:
(i) From equation (28),
g
wrc
¸
∙
1
θ−1
.
= log α
θ (L0 )1−α
(84)
Hence the net competitive real wage is given by
µ
W
P
¶
c
¶
µ
θ−1
1
.
= (1 − t) α
θ (L0 )1−α
(85)
Inspection of this equation reveals that
d
¡W ¢
P
c
dθ
45
> 0.
(86)
(ii) Using (80), (30) and (28) to evaluate (83) for s = θ, and rearranging,
dφ
=
dθ
w
d( Z
(1−β)
Zw )
dθ (1−α)( W )
P
c
n
¡ ¢
(1 − β) W
−
P c
α−β
R
1−β
o
−
d( W
P )
c
dθ
R
2
(1−t)( W
P )
c
D12
nh
(1−β)
(1−α)
i
o
Zw
− 1 Zw + (1 − β)
(87)
Lemma 1:
d(
Zw
Zw
)
dθ
Proof: From (30),
< 0.
Zw
Zu
depends on θ only through Zu and Zu depends on θ only through the
d( θ )
d( θ )
term Dθ . It suffices therefore to show that dθD > 0. But dθD = Dα2 > 0. QED.
¡ ¢
Lemma 2: The participation constraint implies (1 − β) W
− α−β
R > 0.
P c
1−β
Proof: Using equation (15) in (10), the participation constraint can be rewritten as
µ
W
P
¶
>β
c
µ
W
P
¶
+R
(88)
c
or
R
β + ¡ W ¢ ≡ β + r < 1.
P
(89)
c
Simple algebra shows that the statement of the Lemma is equivalent to
β+
α−β
r < 1.
1−β
(90)
Hence, it suffices to establish the inequality in (90). The proof of the Lemma is completed by
noting that, since α < 1, the participation constraint in (89) implies (90).
Consider now the numerator of equation (87). The fact that both α and β are bounded
between zero and one in conjunction with Lemmas 1 and 2 imply that the first term in the
numerator is negative. Equation (86) and the condition α > β, imply that the second term in
the numerator is negative as well. It follows that
dφ
dθ
< 0.
The impacts on unemployment and inflation follow directly from equations (32) and (33).
46
QED
Proof of proposition 3: Equation (30) implies
Zw
=
Zu
θ
(1
D
1−
− n1 )
1
nK
1
+ (1−α)I
K
n
The total derivative of this expression with respect to
d
³
d
Zw
Zu
´
¡1¢ = −
n
1
1 − nK
.
≡
H
1
n
(91)
is given by
(1 − α)αI
< 0.
H 2 KD
(92)
The definitions of N1 and D1 in (80) imply
d (N1 )
(1 − α)αI
¡1¢ = −
H 2 KD
d n
Letting s =
1
n
d (D1 )
1 − β (1 − α)αI
¡1¢ = −
1 − α H 2 KD
d n
(93)
in (83), substituting (92) into (83) and rearranging
d
dφ
¡1¢ = −
n
(1 − β)αI
¡ ¢
(D1 H)2 KD W
P c
½
µ ¶
¾
W
α−β
(1 − β)
−
R .
P c 1−β
(94)
By Lemma 2 the term in curly brackets on the extreme right hand side of this expression is
positive. Since 1 − β > 0 and all the remaining terms are positive,
dφ
1
d( n
)
< 0. That higher
centralization is associated with higher employment and lower inflation follows directly from the
fact that the premium goes down (see (32) and 33). QED
Proof of proposition 4: Differentiating N1 and D1 with respect to t
dD1
R
dN1
¡ ¢ .
=
=
dt
dt
(1 − t) W
P c
47
(95)
Substituting (94) into (83) and rearranging
dφ
R
¡ ¢
= 2
dt
D1 (1 − t) W
P c
∙µ
α−β
1−α
¶
Zw
+1−β
Zu
¸
(96)
Since α ≥ β and α, β < 1 this expression is positive. Hence, an increase in the tax wedge, t,
raises the wage premium, reduces employment (see (32)) and raises inflation (see (33)). QED
Proof of proposition 5: Immediate from inspection of (79) and (80) and by using (32) and
(33). QED
Proof of proposition 6: Letting s = I in (83) and using the definitions in (80) to evaluate
the resulting expression we obtain after some rearrangement
³ ´
½
µ ¶
¾ d Zw
Zu
dφ
1−β
W
α−β
¡ W ¢ (1 − β)
= 2
.
−
R
dI
P c 1−β
dI
D1 (1 − α) P c
(97)
Lemma 2 implies that, given the participation constraint, the expression in large curly parenthesis in (97) is positive. Since the expression preceding it is also positive, the sign of
w
d( Z
Zu )
. Differentiating (91) with respect to I and rearranging
identical to the sign of dI
d
³
Zw
Zu
dI
´
©
ª
(1 − α)(n − 1)
= −©
α + (1 − α)2 DI
ª
2
+ (1 − α)I D
θ(n − 1) K
D
which is unambiguously negative establishing that
dφ
dI
dφ
dI
is
(98)
< 0.
The negative association between u and π on one hand, and I on the other, follows from
(32) and 33). QED
Proof of proposition 7: Since, by propositions 6 and 3
48
dφ
dI
and
dφ
1
d( n
)
are both negative the
statements in the proposition can be established by showing that
d
dt
d
dt
Ã
µ
d
dφ
dI
dφ
¡1¢
n
¶
!
=
=
d2 φ
> 0, and
dIdt
d
d2 φ
¡1¢
n
dt
(99)
> 0.
(i) Differentiating (96) with respect to I and rearranging
2
dφ
= −Q1
dIdt
(
(α − β)
µ
1−β
1−α
¶
³ ´
∙
µ ¶
¸) d Zw
Zu
W
Zw
1
+ ¡ W ¢ (2 − β)2
− (α − β)R
Zu
P c
dI
P c
where Q1 is a positive constant.34 Since 1 − β ≤ 1, Lemma 2 implies that (2 − β)2
¡W ¢
P
c
(100)
− (α −
2
d φ
β)R > 0. Since all the remaining terms in curly parenthesis are also positive, the sign of dIdt
is
Zw
w
d( Z
d
)
(
)
Zu
Zu
opposite to the sign of dI
. But equation (98) implies that dI
< 0 establishing the first
part of (99).
(ii) Differentiating (96) with respect to
2
d
dφ
¡1¢
n
dt
(
= −Q1 (α − β)
µ
1−β
1−α
¶
1
n
and rearranging
³ ´
∙
µ ¶
¸) d Zw
Zu
W
Zw
1
¡1¢ .
+ ¡ W ¢ 2(1 − β)2
− (α − β)R
Zu
P c
d n
P c
(101)
Except for the last terms on their extreme right hand sides, equations (100) and (101) are
d( Zw )
identical. Hence, provided d Z1u < 0, an argument similar to the one used in the proof of part
(n)
d( Zw )
d2 φ
(i) implies that d 1 dt > 0. The proof is completed by noting, from (92), that d Z1u < 0. QED
(n)
(n)
34
Q1 ≡
R
D13 (1−α)(1−t)( W
P )
> 0.
c
49
9.6
9.6.1
Proofs of results underlying the welfare analysis
Derivation of equation (44)
(i) Substituting (75) into (65) and rearranging
α
M
= Ψ5 {1 − φ} 1−α
P
(102)
α
where Ψ5 ≡
γ
1−γ
(1+L0 )α 1−α . θ−1
θ .
θ−1
α2
1
exp 1−α
1−α θ
L0
(ii) Total output, Y, is given by
Y =
Z
0
1
Cja dj
=
Z
1
α
(Lj ) dj =
0
Z
0
1
µ
(L0 (1 − u)) dj = (L0 ) 1 −
α
α
φ
1−α
¶α
(103)
The first equality follows from the continuum of differentiated goods on the zero-one interval, the
second is obtained by using the production function, the third by specializing to a symmetric
equilibrium and the last, by using equation (32). Equation (44) in the text is obtained by
substituting equations (32), (102) and (103) into (43). QED
9.6.2
The impact of φ on b
v(φ)
Differentiating b
v(φ) in (44) with respect to φ and rearranging
£
¤
2α−1
db
v(φ)
α
L0
= −Ψ5
[1 − φ] 1−α −
α (L0 (1 − u))α−1 − R
dφ
1−α
(1 + L0 )(1 − α)
(104)
From equation (32), φ = (1 − α)u implying that, for an internal general equilibrium solution (in
which unemployment is less than one hundred percent), 1 − φ > 0. This in conjunction with the
fact that Ψ5 > 0 implies that the first term on the right hand side of equation (104) is negative.
Condition (45) implies that the second term is also negative. This leads to the following Lemma.
50
de
v(φ)
dφ
Lemma 3:
9.6.3
<0
Proof of proposition 8
Inspection of equation (44) suggests that each of the parameters n1 , t, β, I affects welfare only
through its effect on the wage premium, φ. Hence
db
v(φ)
db
v(φ) dφ
1
=
, s = , t, β, I.
ds
dφ ds
n
Since, by Lemma 3,
the signs of
dφ
.
ds
de
v(φ)
dφ
< 0 the signs of
de
v (φ)
,
ds
(105)
s = n1 , t, β, I are determined by, and opposite, to
The proof of the proposition follows by noting that the signs of
de
v (φ)
,
ds
s = n1 , t, β, I
are given, under the appropriate restrictions (carried over to this proposition) in propositions
3-6. QED
9.7
Government’s choice of tax rate and proof of proposition 10
We start with some preliminaries. Differentiating T in (47) with respect to t,
µ ¶¸
∙
g
dT
L0 1 − α − φ
dφ
Wrc
α·t
=
−
2
dt
1−α
(1 − φ)
(1 − φ)
dt
(106)
g
= exp(−(1 − α)Ψ3 ) ≡ Ψ4 is the gross competitive real wage. Differentiating (106)
where Wrc
with respect to I,
d2 T
αΨ4 L0
≡ H1 =
dt · dI
(1 − α) (1 − φ)2
Since, from proposition 4,
proposition 7,
d2 φ
dt·dI
dφ
dt
¶∙
¸
µ 2 ¶¾
½µ
2t dφ
dφ
dφ
1+
−t
.
−
dI
(1 − φ) dt
dt · dI
(107)
> 0 the first term in curly parenthesis is positive. From part (i) of
> 0. Hence the second term in curly parenthesis is negative. It follows that
the sign of (107) is positive whenever
d2 φ
dt·dI
is not too large.
51
Government’s choice of tax rate and comparative statics on t∗ . The tax rate, t∗ , is
implicitly determined by equation (50) in the text, where
dT
dt
is given by (106) and
de
v
,
dt
obtained
by letting s = t in (105), reduces to:
¤
£
−L0 α [L0 (1 − u)]α−1 − R
db
v
db
v dφ
dφ
=
×
=
×
dt
dφ
dt
(1 + L0 )(1 − α)
dt
(108)
de
v
dφ
in (108), with respect
provided (1 − γ) is sufficiently small. Differentiating the expression for
to I
d2 vb
d
=
dIdφ
dI
"
¤# ∙
£
¸
−L0 α [L0 (1 − u)]α−1 − R
−αLα0 (1 − u)α−2
dφ
=
×
> 0.
(1 + L0 )(1 − α)
(1 + L0 )(1 − α)
dI
(109)
Differentiating (108) with respect to I
From (109) and since
one is negative, since
dφ
dt
de
v
dφ
d2 b
v
dφ db
v
d2 φ
d2 vb
≡ H2 =
×
+
×
dI · dt
dI · dφ
dt
dφ dI · dt
> 0, the first term on the r.h.s. of (110) is positive while the second
< 0 and
d2 φ
dI·dt
> 0.
From the implicit function theorem,
sign of
d2 Υ
dI·dt
(110)
dt∗
dI
=
d2 Υ
dI·dt
2
−d Υ
2
dt
. Hence, the sign of
dt∗
dI
is equal to the
= δ · H1 + (1 − δ) · H2 , where H1 is given by (107), and that of H2 is given by (110).
As we saw above both H1 and H2 are positive when the cross-derivative
Hence, provided the (positive) cross derivative,
proposition 10. QED
52
d2 φ
,
dIdt
d2 φ
dIdt
is sufficiently small,
is sufficiently small.
dt∗
dI
> 0 establishing
9.8
The socially optimal level of conservativeness when fiscal policy
is endogenous
Since welfare depends on CBC only via the wage premium and since
de
v
dφ
< 0, the level of I that
minimizes the wage premium φ also maximizes welfare. Setting β = 0 and (1 − γ) ∼
= 0 in (31),
the total derivative of φ with respect to I can be written as:
⎡"
⎤
# ³ Zw ´
∙
¸ ∗
d
b
Zu
dφ
K ⎣
R
R
Z dt (I) ⎦
¡ W ¢ (1 − α) + α w
=
1 − α¡W ¢
+
dI
1−α
dI
Zu
dI
(1 − t) P c
P c
∙
b
where K is a positive constant, and 1 − α
lated in equation (88). Since
when
dt∗ (I)
dI
≤ 0,
dφ
dI
w
d( Z
Zu )
dI
R
( WP )c
¸
(111)
> 0 by the participation constraint as formu-
< 0, inspection of the right-hand side of (111) reveals that,
is uniformly negative implying that an ultra-conservative central banker is
socially optimal . In this case, the solution for CBC is at a corner and I ∗ = +∞, as was the
case when fiscal policy was exogenous.
By contrast, when
dt∗ (I)
dI
> 0, the socially optimal level of conservativeness
possesses an
µ Z ¶
w
d( Z )
u
internal solution and I ∗ < ∞. This follows from the fact that lim
= 0.
dI
I→∞
10
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11
Definitions of constants
1. θ - elasticity of substitution between any two consumption varieties
2. γ - exponent of consumption aggregator in Dixit-Stiglitz utiliy
3. 1 − γ - exponent of real money balances in Dixit-Stiglitz utiliy
4. α - exponent of labor in production function
5. L0 - number of workers per firm
6. R - value of leisure when unemployed
7. T Rj , j = EW, U W, E - transfer to individual of class j.
8. EW - index for employed worker
9. U W - index for unemployed worker
10. E - index for employer
11. I - central bank conservativeness or independence
12. β - replacement ratio
13. t - tax wedge
14. D ≡ α + θ(1 − α)
15. K ≡ 1 + (1 − α)2 I
16. Zw ≡ 1 −
1
n[1+(1−α)2 I]
> 0;
56
17. Zu ≡
1
n
h
θ(n−1)
α+θ(1−α)
∙ ³
(1−α)I
1+(1−α)2 I
+
¸ Dα
´ 1−α
α
i
´ 1−α
¡ ¢α ³
D
0)
= α1 D γ(1+L
>0
1−γ
¸ Dα
∙
³
´ 1−α
¡ θ ¢ Dα 0
α
γ(1+L0 )
θ
= θ−1
Ψ1 > 0
19. Ψ1 ≡ (θ−1)α
1−γ
´− Dθ ³
´1
³
γ(1+L0 ) D
θ
20. Ψ2 ≡ (θ−1)α
> 0.
1−γ
0
18. Ψ1 ≡
21. Ψm ≡
1
α
γ(1+L0 )
1−γ
l0 −log Ψ2 +[(1−θ)−(1−α)ID] log Ψ2 +(1−α)Ip−1
1+(1−α)2 I
22. Ψ3 ≡ l0 − log Ψ2 +
log Ψ1
1−α
g
23. Ψ4 ≡ exp(−(1 − α)Ψ3 ) = Wrc
α
24. Ψ5 ≡
25. J ≡
γ
1−γ
1
1+L0
(1+L0 )α 1−α . θ−1
θ θ−1
1
α2
exp 1−α
1−α θ
h
L
(θ−1)α
θ
0
α ³
i 1−α
1
g
exp(wrc
)
1
´ 1−α
57